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For everyday experience—together with all sorts of detailed experiments—strongly support the idea that so long as there are no effects from acceleration or external forces, physical systems work exactly the same regardless of how fast they are moving.
The overall structure of space that emerges may be complicated, and there may be objects that end up moving at all sorts of speeds.
And certainly this allows us with rather modest effort to do quite well in handling all sorts of data that we choose to interact with in everyday life.
But as soon as one allows more than two possible colors, or allows dependence on more than just nearest neighbors, one immediately finds all sorts of further examples of class 4 behavior.
So the result is that computational irreducibility can in the end be expected to be common, so that it should indeed be effectively impossible to outrun the evolution of all sorts of systems.
And indeed there are all sorts of well-known examples—such as Fermat's Last Theorem and the Four-Color Theorem—in which a theorem that is easy to state seems to require a proof that is immensely long.
But what the pictures on the next two pages [ 804 , 805 ] show is that the vast majority of axiom systems actually allow operators with all sorts of different forms.
But one might have assumed that to achieve their universality these axiom systems would have to be specially set up with all sorts of specific sophisticated features.
And the point is that if computational irreducibility is present, then there is in a sense all sorts of information about the behavior of a system that can only be found from its rules by doing an irreducibly large amount of computational work.
But the discoveries in this book have made it clear that in fact such computation is quite common in all sorts of systems that do not show anything that we would normally consider intelligence.